What is the "thing" which is trained in AI model training
This answer applies to Machine Learning (ML) part of AI, as that seems to be what you are asking about. Please bear in mind that AI is still a broad church, including many other techniques than ML. ML, including neural networks for deep learning, and Reinforcement Learning (RL) is only a subset of AI - some AI techniques are more focused on the algorithm than parameters.


  
  What are the trained models? are they algorithms or a collection of parameters in a file?
  


In ML, the usual process is to feed data into parametric function (e.g. a neural network) and alter the parameters of it to "fit" the data. The main output of this is a collection of parameters and hyperparameters that describe the parametric function. So 90% of the time, when discussing the "trained model", it means the same thing as the collection of paramaters.

However, those parameters are of limited use without a library that can re-create the function from them. Parameters will be saved from a specific library and can be loaded back into that library easily. It is also possible for libraries to read or convert from models saved from other libraries, much like how different spreadsheet programs can read each others' files.


  
  What do they look like? e.g. file extensions
  


This varies a lot, depending on which library was used. It is not possible to make a general statement. For example, Tensorflow can save variables to a "Checkpoint" file with .ckpt extension, but can get more sophisticated depending on how much of the model you want to export, and full models with whole structure will contain more than just the variables and have the .pb extension.


  
  Especially, I want to find the trained models for detecting birds (the bird types do not matter). Are there any platforms for open-source/free online trained AI models??
  


There are a few places where you can find selections of pre-trained models. One such place is Tensorflow's Model Zoo and you might be interested in Tensorflow detection model zoo. 

Other frameworks may also provide example code. For instance Caffe also has a "Model Zoo" (searching Model Zoo is a good starting strategy).

If you are working at the level of collecting model parameters and want to run these models yourself, you will need to learn a bit about each library, what language is used to work with it, and maybe follow some tutorial about how to use it. A few models will be packaged up with working scripts to use from the command line, but many are not and may take sometime and effort to get working.

When you have a specific detection target, you may be disappointed to find models that don't quite match what you want. For image classifying, it is common if you have a specialist need to take a pre-existing general model that has been trained on large dataset for weeks, and then "fine tune" it with your own image dataset for your purpose. Most NN libraries will have tutorials and examples of this fine tuning process.